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Number · Standard form

Chapter 1 · 4

The idea

Calculating in standard form

Why multiplying and dividing in standard form is just the index laws, why the answer often needs re-standardising, and why adding needs the powers aligned first.

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Number · Standard form

Calculating in standard form

Why multiplying and dividing in standard form is just the index laws, why the answer often needs re-standardising, and why adding needs the powers aligned first.

Why it works

The index laws in a lab coat

Standard-form arithmetic is the index laws wearing a lab coat. Multiplication can be regrouped freely, so pair up the parts:

(3×104)×(2×105)=6×109(3 \times 10^4) \times (2 \times 10^5) = 6 \times 10^9

Division the same way: coefficients divide, powers subtract —

8×1073.2×102=83.2×107−2=2.5×105.\frac{8 \times 10^7}{3.2 \times 10^2} = \frac{8}{3.2} \times 10^{7-2} = 2.5 \times 10^5.

Re-standardise the answer

The arithmetic is oblivious to the 1≤a<101 \le a < 10 rule, so it often lands outside it:

(4×105)×(5×103)=20×108.(4 \times 10^5) \times (5 \times 10^3) = 20 \times 10^8.

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